Topology and geometry of chain, ribbon and tube silicates: generation and analysis of infinite one-dimensional arrangements of (TO4)n- tetrahedra
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Date
2022-10-04
Authors
Day, Maxwell C.
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Abstract
A structure hierarchy for chain, ribbon and tube silicate minerals is proposed. Chains of (SiO4)4- tetrahedra are described topologically as chain graphs, where tetrahedra are represented as vertices and the linkages between them as edges. Chain arrangements are organized into classes and sub-classes based the number and connectivity of tetrahedra or vertices in the repeat unit (unit cell) of each chain arrangement. Chain graphs that correspond to the most abundant minerals have O:T = 3:1 - 2.75:1 and chains with O:T < 2.5:1 – 2:1 are not observed in minerals despite being topologically possible.
A graph-theoretical method is proposed for converting each infinite chain graph to a finite wrapped graph such that it can be described using an adjacency matrix. A method (MatLab script) for generating all possible non-isomorphic chain graphs (up to a boundary condition, ∑r = 8) is proposed. Approximately 1500 non-isomorphic chain graphs are generated, ~50 of which are observed in chain silicate minerals. A software program (GraphT-T) is given for embedding chain graphs in Euclidean space to gauge their geometrical compatibility with the observed metrics of crystal structures. The average distance between linked T-cations (T-T distances) and the minimum distance between unlinked T-cations (T…T separations) were determined for all chain-silicate minerals to be 3.06 ± 0.15 Å and 3.71 Å, respectively. These values are used to constrain the geometry of chain graphs once embedded. If the resultant chain graphs have geometries that satisfy the T-T and T…T constraints, they are compatible with the metrics of crystal structures and may occur; if they do not, they are incompatible and are unlikely to occur.
As the average vertex connectivity (1-4) increases, the e/n (edge/vertex) ratio increases and all chains with e/n < 1.5 are compatible and many chains with e/n > 1.5 are incompatible. This compatibility change at e/n = 1.5 coincides with decrease in chain flexibility and many chains with e/n ≥ 1.5 require distortion to satisfy the T-T and T…T constraints. As e/n = 1.5 corresponds to an O:T = 2.5, the above observations explain why chains with O:T = 2.5 – 2.0 are not observed in minerals.
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Keywords
Silicate minerals, (SiO4)4- tetrahedra, Chains of tetrahedra, Structure hierarchy, Infinite graphs, Topology, Graph theory, Crystal structures, Mineral stability, Graph embeddings